ISSN:

1930-8337

eISSN:

1930-8345

## Inverse Problems & Imaging

2013 , Volume 7 , Issue 1

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2013, 7(1): 1-25
doi: 10.3934/ipi.2013.7.1

*+*[Abstract](388)*+*[PDF](4059.5KB)**Abstract:**

In this paper we construct a shape space of medial ball representations from given shape training data using methods of Computational Geometry and Statistics. The ultimate goal is to employ the shape space as prior information in supervised segmentation algorithms for complex geometries in 3D voxel data. For this purpose, a novel representation of the shape space (i.e., medial ball representation) is worked out and its implications on the whole segmentation pipeline are studied. Such algorithms have wide applications for industrial processes and medical imaging, when data are recorded under varying illumination conditions, are corrupted with high noise or are occluded.

2013, 7(1): 27-46
doi: 10.3934/ipi.2013.7.27

*+*[Abstract](271)*+*[PDF](389.3KB)**Abstract:**

It has been shown in [10] that on a simple, compact Riemannian 2-manifold the attenuated geodesic ray transform, with attenuation given by a connection and Higgs field, is injective on functions and 1-forms modulo the natural obstruction. Furthermore, the scattering relation determines the connection and Higgs field modulo a gauge transformation. We extend the results obtained therein to the case of magnetic geodesics. In addition, we provide an application to tensor tomography in the magnetic setting, along the lines of [11].

2013, 7(1): 47-79
doi: 10.3934/ipi.2013.7.47

*+*[Abstract](502)*+*[PDF](4753.9KB)**Abstract:**

In order to prevent influx of highly enriched nuclear material throu-gh border checkpoints, new advanced detection schemes need to be developed. Typical issues faced in this context are sources with very low emission against a dominating natural background radiation. Sources are expected to be small and shielded and hence cannot be detected from measurements of radiation levels alone. We consider collimated and Compton-type measurements and propose a detection method that relies on the geometric singularity of small sources to distinguish them from the more uniform background. The method is characterized by high sensitivity and specificity and allows for assigning confidence probabilities of detection. The validity of our approach can be justified using properties of related techniques from medical imaging. Results of numerical simulations are presented for collimated and Compton-type measurements. The 2D case is considered in detail.

2013, 7(1): 81-105
doi: 10.3934/ipi.2013.7.81

*+*[Abstract](364)*+*[PDF](939.5KB)**Abstract:**

The full application of Bayesian inference to inverse problems requires exploration of a posterior distribution that typically does not possess a standard form. In this context, Markov chain Monte Carlo (MCMC) methods are often used. These methods require many evaluations of a computationally intensive forward model to produce the equivalent of one independent sample from the posterior. We consider applications in which approximate forward models at multiple resolution levels are available, each endowed with a probabilistic error estimate. These situations occur, for example, when the forward model involves Monte Carlo integration. We present a novel MCMC method called $MC^3$ that uses low-resolution forward models to approximate draws from a posterior distribution built with the high-resolution forward model. The acceptance ratio is estimated with some statistical error; then a confidence interval for the true acceptance ratio is found, and acceptance is performed correctly with some confidence. The high-resolution models are rarely run and a significant speed up is achieved.

Our multiple-resolution forward models themselves are built around a new importance sampling scheme that allows Monte Carlo forward models to be used efficiently in inverse problems. The method is used to solve an inverse transport problem that finds applications in atmospheric remote sensing. We present a path-recycling methodology to efficiently vary parameters in the transport equation. The forward transport equation is solved by a Monte Carlo method that is amenable to the use of $MC^3$ to solve the inverse transport problem using a Bayesian formalism.

2013, 7(1): 107-122
doi: 10.3934/ipi.2013.7.107

*+*[Abstract](353)*+*[PDF](332.6KB)**Abstract:**

This article is concerned with the representation of curves by means of integral invariants. In contrast to the classical differential invariants they have the advantage of being less sensitive with respect to noise. The integral invariant most common in use is the circular integral invariant. A major drawback of this curve descriptor, however, is the absence of any uniqueness result for this representation. This article serves as a contribution towards closing this gap by showing that the circular integral invariant is injective in a neighbourhood of the circle. In addition, we provide a stability estimate valid on this neighbourhood. The proof is an application of Riesz--Schauder theory and the implicit function theorem in a Banach space setting.

2013, 7(1): 123-157
doi: 10.3934/ipi.2013.7.123

*+*[Abstract](295)*+*[PDF](1415.4KB)**Abstract:**

The aim of our work is to reconstruct an inclusion $\omega$ immersed in a fluid flowing in a larger bounded domain $\Omega$ via a boundary measurement on $\partial\Omega$. Here the fluid motion is assumed to be governed by the Stokes equations. We study the inverse problem of reconstructing $\omega$ thanks to the tools of shape optimization by minimizing a Kohn-Vogelius type cost functional. We first characterize the gradient of this cost functional in order to make a numerical resolution. Then, in order to study the stability of this problem, we give the expression of the shape Hessian. We show the compactness of the Riesz operator corresponding to this shape Hessian at a critical point which explains why the inverse problem is ill-posed. Therefore we need some regularization methods to solve numerically this problem. We illustrate those general results by some explicit calculus of the shape Hessian in some particular geometries. In particular, we solve explicitly the Stokes equations in a concentric annulus. Finally, we present some numerical simulations using a parametric method.

2013, 7(1): 159-182
doi: 10.3934/ipi.2013.7.159

*+*[Abstract](311)*+*[PDF](479.0KB)**Abstract:**

We study the inverse problem of the simultaneous identification of two discontinuous diffusion coefficients for a one-dimensional coupled parabolic system with the observation of only one component. The stability result for the diffusion coefficients is obtained by a Carleman-type estimate. Results from numerical experiments in the one-dimensional case are reported, suggesting that the method makes possible to recover discontinuous diffusion coefficients.

2013, 7(1): 183-197
doi: 10.3934/ipi.2013.7.183

*+*[Abstract](273)*+*[PDF](373.1KB)**Abstract:**

We investigate two inverse scattering problems for the nonlinear Schrödinger equation $$ -\Delta u(x) + h(x,|u(x)|)u(x) = k^{2}u(x), \quad x \in \mathbb{R}^2, $$ where $h$ is a very general and possibly singular combination of potentials. The method of Born approximation is applied for the recovery of local singularities and jumps from fixed angle scattering and backscattering data.

2013, 7(1): 199-216
doi: 10.3934/ipi.2013.7.199

*+*[Abstract](299)*+*[PDF](522.8KB)**Abstract:**

In this paper we integrate the SART (Simultaneous Algebraic Reconstruction Technique) algorithm into a general iterative method, introduced in [8]. This general method offers us the possibility of achieving a new convergence proof of the SART method and prove the convergence of the constrained version of SART. Systematic numerical experiments, comparing SART and Kaczmarz-like algorithms, are made on two phantoms widely used in image reconstruction literature.

2013, 7(1): 217-242
doi: 10.3934/ipi.2013.7.217

*+*[Abstract](401)*+*[PDF](1839.1KB)**Abstract:**

Electrical impedance tomography (EIT) aims to reconstruct the electric conductivity inside a physical body from current-to-voltage measurements at the boundary of the body. In practical EIT one often lacks exact knowledge of the domain boundary, and inaccurate modeling of the boundary causes artifacts in the reconstructions. A novel method is presented for recovering the boundary shape and an isotropic conductivity from EIT data. The first step is to determine the minimally anisotropic conductivity in a model domain reproducing the measured EIT data. Second, a Beltrami equation is solved, providing shape-deforming reconstruction. The algorithm is applied to simulated noisy data from a realistic electrode model, demonstrating that approximate recovery of the boundary shape and conductivity is feasible.

2013, 7(1): 243-252
doi: 10.3934/ipi.2013.7.243

*+*[Abstract](237)*+*[PDF](325.4KB)**Abstract:**

We study the spherical mean transform on $\mathbb{R}^n$. The transform is characterized by the Euler-Poisson-Darboux equation. By looking at the spherical harmonic expansions, we obtain a system of $1+1$-dimension hyperbolic equations. Using these equations, we discuss two known problems. The first one is a local uniqueness problem investigated by M. Agranovsky and P. Kuchment, [

*Memoirs on Differential Equations and Mathematical Physics*,

**52**(2011), 1--16]. We present a proof which only involves simple energy arguments. The second problem is to characterize the kernel of spherical mean transform on annular regions, which was studied by C. Epstein and B. Kleiner [

*Comm. Pure Appl. Math.*,

**46(3)**(1993), 441--451]. We present a short proof that simultaneously provides the necessity and sufficiency for the characterization. As a consequence, we derive a reconstruction procedure for the transform with additional interior (or exterior) information.

We also discuss how the approach works for the hyperbolic and spherical spaces.

2013, 7(1): 253-265
doi: 10.3934/ipi.2013.7.253

*+*[Abstract](248)*+*[PDF](325.2KB)**Abstract:**

Photoacoustic tomography is a rapidly developing medical imaging technique that combines optical and ultrasound imaging to exploit the high contrast and high resolution of the respective individual modalities. Mathematically, photoacoustic tomography is divided into two steps. In the first step, one solves an inverse problem for the wave equation to determine how tissue absorbs light as a result of a boundary illumination. The second step is generally modeled by either diffusion or transport equations, and involves recovering the optical properties of the region being imaged.

In this paper we, address the second step of photoacoustics, and in particular, we show that the absorption coefficient in the stationary transport equation can be recovered given certain internal information about the solution. We will consider the variable index of refraction case, which will correspond to an inverse transport problem on a Riemannian manifold with internal data and a known metric. We will prove a stability estimate for a functional of the absorption coefficient of the medium by finding a singular decomposition for the distribution kernel of the measurement operator. Finally, we will use this estimate to recover the desired absorption properties.

2013, 7(1): 267-282
doi: 10.3934/ipi.2013.7.267

*+*[Abstract](303)*+*[PDF](1846.2KB)**Abstract:**

The X-ray phase contrast imaging technique relies on the measurement of the Fresnel diffraction intensity patterns associated to a phase shift induced by the object. The simultaneous recovery of the phase and of the absorption is an ill-posed nonlinear inverse problem. In this work, we investigate the resolution of this problem with nonlinear Tikhonov regularization and with a joint sparsity constraint regularization. The regularization functionals are minimized with a Gauss-Newton method and with a fixed point iterative method based on a surrogate functional. The algorithms are evalutated using simulated noisy data. The joint sparsity regularization gives better reconstructions for high noise levels.

2013, 7(1): 283-290
doi: 10.3934/ipi.2013.7.283

*+*[Abstract](249)*+*[PDF](757.5KB)**Abstract:**

Source extraction in audio is an important problem in the study of blind source separation (BSS) with many practical applications. It is a challenging problem when the foreground sources to be extracted are weak compared to the background sources. Traditional techniques often do not work in this setting. In this paper we propose a novel technique for extracting foreground sources. This is achieved by an interval of silence for the foreground sources. Using this silence interval one can learn the background information, allowing the removal or suppression of background sources. Very effective optimization schemes are proposed for the case of two sources and two mixtures.

2013, 7(1): 291-303
doi: 10.3934/ipi.2013.7.291

*+*[Abstract](375)*+*[PDF](330.1KB)**Abstract:**

We consider an interior inverse scattering problem of reconstructing the shape of a cavity. The measurements are the scattered fields on a curve inside the cavity due to one point source. We employ the decomposition method to reconstruct the cavity and present some convergence results. Numerical examples are provided to show the viability of the method.

2013, 7(1): 305-306
doi: 10.3934/ipi.2013.7.305

*+*[Abstract](241)*+*[PDF](217.6KB)**Abstract:**

In paper "Strongly Convex Programming for Exact Matrix Completion and Robust Principal Component Analysis", an explicit lower bound of $\tau$ is strongly based on Theorem 3.4. However, a coefficient is missing in the proof of Theorem 3.4, which leads to improper result. In this paper, we correct this error and provide the right bound of $\tau$.

2016 Impact Factor: 1.094

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